Lawvere theory
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In category theory, a Lawvere theory (named after American mathematician William Lawvere) is a category that can be considered a categorical counterpart of the notion of an equational theory. Intuitively, it is a categorical generalization of algebraic structures (e.g., a group or a ring), where there exists a "generic" object and all objects are isomorphic to an integer power of , representing the inputs for the -ary operations on (i.e., of the form , starting from the fact that and so ; trivially generalizing inductively, we get the rest of the objects) where the operations come from the algebraic structure at hand (e.g., addition and/or multiplication).
The Lawvere theory of groups has as its generic object an underlying placeholder where the other objects are the inputs for -ary operations from those integer powers of of the form () back to , where a model, a finite-product preserving functor, from this theory into a target category such as the category of sets or topological spaces, would map the abstract theory onto the category to create a concrete, "combined," group-based structure. This model would provide a set with group structure (a group) or a topological space with group structure (a topological group), supplying appropriate names to the generic object and its mappings (-ary operations) according to whatever the theory and model at play are; in the model of sets for the Lawvere theory of groups, the generic object is a group and its mappings are group operations.
Definition
Let be a skeleton of the category FinSet of finite sets and functions. Formally, a Lawvere theory consists of a small category with (strictly associative) finite products and a strict identity-on-objects functor preserving finite products.
A model of a Lawvere theory in a category with finite products is a finite-product preserving functor . A morphism of models where and are models of is a natural transformation of functors.
Model examples
Some examples of models of the Lawvere theory of groups (i.e., = ):
- ( is a classical group)
- ( is a topological group)
- ( is a Lie group).
Some examples of models of the Lawvere theory of rings (i.e., = ):
- ( is a classical ring)
- ( is a topological ring)
- ( is a smooth ring).
Category of Lawvere theories
A map between Lawvere theories and is a finite-product preserving functor that commutes with and . Such a map is commonly seen as an interpretation of in .
More formally, a map , such that , where and .
Lawvere theories together with maps between them form the category .
Variations
Variations include multisorted (or multityped) Lawvere theory, infinitary Lawvere theory, and finite-product theory.[1]